Towards the classification of homogeneous third-order Hamiltonian operators

TL;DR

This paper classifies third-order Hamiltonian operators of differential-geometric type using algebraic geometry, providing a parametrization and classification for dimensions up to four.

Contribution

It introduces a parametrization of these operators via algebraic varieties and classifies them for dimensions up to four.

Findings

Operators correspond to elements of rank n in a specific algebraic variety.

Classification achieved for n ≤ 4.

Different orbits under SL(n+1) correspond to non-equivalent operators.

Abstract

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(\Lambda^2V)$ that lie in the kernel of the natural map $S^2(\Lambda^2V)\to \Lambda^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $n\leq 4$.